Competing with Loyalty Discounts

by

Patrick Greenlee and David Reitman*

February 4, 2005

Abstract

Loyalty discounts, offered to customers that meet purchase thresholds, can shift share

from rival firms. In a differentiated product duopoly, only one firm employs a program

that customers adopt in equilibrium. Whenever consumers strongly prefer the product of

said firm, such discounts increase producer surplus. When a firm requires customers to

meet thresholds in multiple rivalrous markets, the loyalty discount and consumer surplus

may fall. Finally, when a “branded” product monopolist faces competition for an

unrelated “generic” product, the monopolist raises the branded spot price, and offers a

discount to customers that purchase generic product only from the monopolist.

* U.S. Department of Justice, patrick.greenlee@usdoj.gov, and david.reitman@usdoj.gov. This is

a revised version of EAG Discussion Paper 04-2. We have benefited from discussions with Ken

Elzinga, Chris Hardee, Sheldon Kimmel, Dan O’Brien, David Sibley, Mary Sullivan, and Greg

Werden. The analysis and conclusions herein are solely those of the authors and do not represent

the views of the United States Department of Justice.

1

1. Introduction

When consumers make multiple purchases, sellers have an incentive to employ

non-linear pricing strategies in order to increase sales volume while simultaneously

extracting greater surplus on inframarginal units. Like volume discounts and two-

part tariffs, loyalty discounts can be considered another instrument in the same

pricing toolbox: a reward in the form of lower prices to customers who meet a

purchase threshold.

Loyalty discounts can refer to something as simple as a coffee shop offering a

free cup of coffee for every ten cups purchased. The term “loyalty discount,” however,

typically applies to pricing strategies that condition a discount on satisfying a

minimum purchase requirement in each period. The target threshold may be based

on purchase volume, growth over the previous period, or share of purchase

requirements. While this definition does not clearly differentiate loyalty discounts

from volume discounts or other forms of non-linear pricing, several elements

distinguish loyalty discounts. First, loyalty discounts typically require allocating a

substantial share of total purchases to a single supplier. The term “market share

discount” is often employed to describe this pricing strategy. Second, loyalty

programs separate customers into two distinct groups: those who qualify, and those

who do not, unlike the more gradual gradations possible with other pricing

strategies. In addition, because the target is often not based solely on current

purchases from a given seller, two customers that purchase the same volume from a

seller may pay different prices.1

Even in the simple case of an occasional free cup of coffee for regular patrons,

the implications of a loyalty discount scheme may not be as straightforward as at

first glance: do regular customers really enjoy a discount, or do other customers pay

a non-loyalty surcharge? In either case, this pricing scheme can be classified as a

form of price discrimination, with firms competing more aggressively for loyal

customers. Combining the discount with a minimum purchase requirement in each 1 This discriminatory aspect has contributed to the abuse of dominance scrutiny that loyalty

rebate schemes have attracted, particularly in the European Community; see for example,

Case No. IV/D-2/34.780, Virgin/British Airways (1999).

2

period complicates the analysis. While the same elements of aggressive pricing and

price discrimination are present, there is now an additional question about whether

these purchase requirements, coupled with a loyalty discount for buyers who comply

with the purchase terms, can function as exclusionary behavior to the detriment of

rival firms and competition. This is of particular concern when the firm offering

loyalty discounts is much larger than its rivals. While the discount on each unit sold

may be small, the aggregate loyalty discount across all products may generate a

substantial rebate. A smaller competitor, able only to offer a larger discount on its

smaller sales volume, may find it unprofitable to match the overall discount. Thus

such discounts can be a powerful tool in shifting sales to the larger firm.

Several recent antitrust cases have brought the potential exclusionary role of

loyalty discounts to the forefront. In Concord Boat, the dominant manufacturer of

stern drive marine engines, Brunswick, offered discounts of several percent to

buyers that bought a minimum share of their engines from Brunswick.2 A number of

boat builders jointly won a jury verdict in a suit against Brunswick, but lost on

appeal. In Virgin Atlantic, British Airways offered loyalty discounts to travel agents

and corporate customers that satisfied minimum purchase requirements across a

number of routes served by British Airways out of Heathrow Airport.3 Virgin

Atlantic sued, claiming that these discounts contributed to “predatory foreclosure” of

Virgin from several Heathrow routes. British Airways won a summary judgment

motion, which was upheld on appeal, on the grounds that Virgin had not offered

adequate proof of the predatory claim. While British Airways successfully defended

its discount policies in American courts, the European Commission found that its

commission programs violated Article 82. In a third case, 3M was accused of using

loyalty discounts across a number of different products to unlawfully exclude

LePage’s in the market for transparent tape.4 The decision has gone back in forth in

the courts, with the most recent en banc opinion of the Third Circuit going against

3M. In June 2004, the Supreme Court declined to hear the case.

2 Concord Boat Corp. v. Brunswick Corp., 207 F.3d 1039 (8th Cir. 2000).

3 Virgin Atlantic Airways Ltd. V. British Airways PLC, 257 F.3d 256 (3rd Cir. 2001); Case No.

IV/D-2/34.780, Virgin/British Airways (1999).

4 LePage’s Inc. v. 3M Co. 324 F.3d 141 (3rd Cir 2003), cert. denied, 124 S.Ct. 2932 (2004).

3

The shifting outcomes of these cases and the legal commentary that they have

spawned suggest that whether loyalty discounts are anti-competitive depends

greatly on the legal framework under which they are analyzed. Is the issue whether

the discounts effectively foreclose competitors from a sufficient share of the market?

Is it whether the discounts lead to sales below cost, as was alleged by the plaintiff in

Virgin Atlantic, and was the decisive issue in Concord Boat, and in the decision by

the Third Circuit panel in LePage’s? Does it matter whether the intent of the

dominant firm was exclusionary, or whether there is harm to consumers?5

One element absent from the debate is an understanding of when firms would

use loyalty programs as a competitive strategy, even in the absence of exclusionary

motives. This paper contributes by studying the equilibrium use of loyalty programs,

and how they affect pricing, market shares and welfare. In addition to providing a

better backdrop against which to observe exclusionary or predatory behavior, our

analysis lays the groundwork for tests that can distinguish such behavior.

The incentive to employ loyalty discounts as a competitive strategy in a

differentiated product market is similar to the incentive to use non-linear pricing

(e.g. frequent flyer programs) more generally. Stole (1995), Hamilton and Thisse

(1997), Gasmi, Moreaux, and Sharkey (2000), Mark and Vickers (2001), Min et al.

(2002), and Page and Montiero (2003), among others, have shown that non-linear

pricing typically emerges as an equilibrium strategy in a price setting oligopoly.6

Loyalty programs make marginal purchases less price sensitive, which is similar to

imposing switching costs.7 A portion of the literature treats loyalty discounts as a

form of partial exclusive dealing that can generate pro-competitive and anti-

competitive effects. On the pro-competitive side, Mills (2004) shows how market

share discounts can lessen the incentive to free ride on manufacturer supplied

promotions. In contrast, Tom, Balto, and Averitt (2000) discuss potential

exclusionary effects of market share discounts, while Cairns and Galbraith (1990)

and Larsen and Storm (2002) identify circumstances under which frequent flyer

5 O’Donoghue (2002), Keyte (2003), and Hewitt (2003) discuss these different perspectives.

6 Kolay, Shaffer, and Ordover (2004) study a monopolist’s use of similar all-units discounts.

7 See, for example, Caminal and Matutes (1990) and the survey by Klemperer (1995).

4

discounts can create entry barriers. Finally, some research addresses multi-product

loyalty programs as a form of tying. Greenlee, Reitman, and Sibley (2004), and

Nalebuff (2004a,b) examine how a monopolist’s loyalty program can leverage market

power into competitive markets.

As the variety of legal cases discussed above suggests, no single model covers all

market structures that raise concern about the anti-competitive use of loyalty

discounts. In cases like Concord Boat, the concern can stem from the use of loyalty

discounts by a large firm in a single market. The potential for abuse may be greater

if the dominant firm offers a loyalty program that links multiple markets, as in

Virgin Atlantic. And finally, a monopolist in one market may use loyalty discounts to

extend its market power to other markets, as was alleged in LePage’s. This paper

presents a set of models that covers this range of market structures.

This paper is organized in the following manner. Section 2 introduces a duopoly

model of a single market, like Concord Boat, in which firms offer loyalty contracts

based on sales in that market. Firms have identical costs but may differ in the value

customers place on their products. In this setting, firms always choose to use a

loyalty program in equilibrium, aside from any exclusionary motive, but only one

firm’s program is adopted by consumers. Typically, the firm selling the preferred

product offers the accepted loyalty discount, and such discounts increase producer

surplus whenever preferences for said firm’s product are sufficiently strong.

Section 3 extends the model to settings, like Virgin Atlantic, in which a large

firm can link the discount across a number of related markets. When some rivals

cannot offer loyalty programs, linking several rivalrous markets in a loyalty

program allows the large firm to reduce the discount. The analysis is followed by a

discussion that distinguishes the equilibrium strategy in the model from intentional

exclusionary behavior, and offers a test to distinguish between the two.

Section 4 introduces a second market in which one firm is a monopolist, and

asks whether the monopolist has an incentive to use loyalty discounts to link sales

between the monopolized market and a rivalrous market. Again, even without an

exclusionary motive, the answer typically is yes. As in Section 3, the analysis is

5

followed by a discussion of when loyalty discounts across monopolized and rivalrous

markets may be exclusionary. Section 5 offers a brief conclusion. An Appendix

contains the proofs of our first and final propositions.

2. Loyalty discounts in a single market

These models incorporate some of the important common elements of loyalty

programs. While loyalty discounts may apply only to incremental sales, one common

feature of the cases discussed above is that the discounts applied to every unit sold.8

Our models assume that firms offer discounts on all units sold whenever a customer

satisfies the loyalty requirement. A spot price pertains to customers who, perhaps

due to size or an inability to be readily monitored, are not candidates for a loyalty

program. In addition to spot prices, firms select the prices and restrictions that

characterize a loyalty program. Customers then choose among the available prices

and programs. By assumption there is foresight on both sides: customers know at

the outset whether they will fulfill the terms of each program, and firms correctly

anticipate product choices and customers’ success in meeting program requirements.

Consider two firms, A and B, that each sell a differentiated product and compete

by setting prices. Both firms have constant marginal costs c . The firms sell to two

groups of customers. Large customers buy multiple products, and for each purchase

the customer has a different evaluation of the relative value of firm A’s and firm B’s

products. Large customers, for example, may be travel agents, and each purchase

may reflect the idiosyncratic preferences of an individual client. Small customers

make individual purchases at the spot price, and are not candidates for a loyalty

program. Let 0θ > be the ratio of purchases made by small customers to purchases

made by large customers, so that the fraction of all purchases that are made by large

customers is ).1/(1 θ+

An important feature of these models is that they incorporate two forms of

customer heterogeneity. One, as mentioned above, is that some customers are not

candidates for the loyalty program and purchase only at the spot price. Without

these customers, firms could use the spot price as an out-of-equilibrium threat: join

8 Such price reductions are often referred to as “dollar-one,” “all-unit,” or “rollback” discounts.

6

my loyalty program or pay an exorbitant price.9 The first model also assumes that

the products of different firms are imperfect substitutes. Customers make multiple

purchases, but each purchase is idiosyncratic in the extent to which it is better

suited for one or the other firm’s product. This assumption makes it costly, both to

the firm and to social welfare, to shift many purchases to one firm via a loyalty

program. The pecuniary benefits to firms, however, typically exceed this loss.

Small and large customers have the same distribution of reservation prices across

purchases, though they may have a different total number of purchases. Purchases

of firm B’s product have reservation price, 2/xR + , while the reservation price for

firm A’s product is 2/xR − , where x is distributed according to )(xF on [ ],x L H∈ .

We assume the corresponding density function )(xf is strictly positive on [ ]HL, .

The mix of purchases is assumed to be deterministic, so that large customers can

ensure that they purchase a σ share of their purchases from firm A by allocating all

purchases with 1( )x F σ−≤ to firm A.10 We maintain the following monotone

hazard assumption in order to guarantee that both firms have the usual shaped

differentiated product Bertrand reaction functions, with slope less than one, so that

this equilibrium exists and is unique.

ASSUMPTION 1: For all ],,[ HLx ∈ .)(

)(10)()(

−>>

xf

xFdxd

xfxF

dxd

In general, demand uncertainty may prevent customers from perfectly

predicting their own aggregate demand or how that demand breaks out among

available brands. Nevertheless, customers often take steps to ensure that they hit a

target by shifting purchases across accounting periods, by steering purchases to a

particular brand, or by buying for inventory. Such steps provide greater assurance

that purchasers will clear program hurdles. Accordingly, the models in the present

9 The effect of this type of out-of-equilibrium threat can still be determined by considering

the limiting case where the number of non-participating customers is vanishingly small.

10 Alternatively, one can assume that individual purchases are atomistic, so that, by the law

of large numbers, the program is satisfied with probability one using the same decision rule.

7

analysis downplay demand uncertainty. We assume throughout that customers

achieve the program target for a supplier with certainty by allocating sufficient sales

to that supplier’s product. Given the absence of demand uncertainty, we downplay

the type of sales target employed. Customers may prefer share targets to volume

targets, for example, when the distribution of demand across brands is more

predictable than the overall level of demand. Conversely, a volume target may be

better when demand for the dominant firm’s product is more predictable than

demand for competing products. The choice of levels or growth rates depends in part

on whether demand shocks are correlated across time. The models here assume that

firms offer a discount on all purchases if customers meet a designated market share

target, but an equivalent result holds when firms use volume or growth targets.11

Firms simultaneously set a spot price, ip , and a loyalty program, },{ ii qz , that

offers a discounted price iz to large customers that satisfy firm i ’s share

requirement. Firm A charges Az for each unit whenever the customer buys at least

an )( AqF share of its purchases from firm A. Similarly, firm B charges Bz for each

unit whenever the customer buys at least a )(1 BqF− share from firm B. After prices

and loyalty programs have been announced, customers decide whether to satisfy

either program and allocate purchases accordingly.

For small customers and large customers that fail to meet either program’s

terms, the optimal decision allocates purchases to the firm offering the higher

surplus, given the announced spot prices. For each purchase, customers receive a

surplus of ApxR −− 2/ whenever they buy from firm A and BpxR −+ 2/ whenever

they buy from firm B. Thus these customers buy from firm A whenever AB ppx −<

and from firm B whenever .AB ppx −>

If customers satisfy firm A’s loyalty program but not firm B’s, and if firm A’s

contract is a binding constraint for customers that satisfy it, then firm profits are:

11 For the same reason, we do not distinguish between the terms loyalty discounts, loyalty

rebates, fidelity rebates, and market share discounts, which refer to similar pricing schemes.

8

({ , , }, ) ( ) ( ) ( ) ( )({ , , }, ) ( )[1 ( ) ( )].

andA A A A B A A A B A

B A A A B B A B A

z q p p z c F q p c F p pz q p p p c F q F p p

θθ θ

Π = − + − −Π = − − + − −

(1)

Alternatively, if customers only satisfy firm B’s program and this program is a

binding constraint for customers that satisfy it, then firm profits are:

)].(1)[()](1[)(}),,{,(and)]()()[(}),,{,(

ABBBBBBBAB

ABBABBBAA

ppFcpqFczpqzpppFqFcppqzp

−−−+−−=Π−+−=Π

θθ

(2)

Finally, if customers opt for neither program, then profits are

)].(1[)1()(),(and)()1()(),(

ABBBAB

ABABAA

ppFcpppppFcppp

−−+−=Π−+−=Π

θθ

(3)

For this latter case, the equilibrium spot prices are the prices in the differentiated

product Bertrand equilibrium of the game without loyalty programs. Those prices

satisfy the first order conditions:

* * * *

* ** * * *

( ) 1 ( )( ) ( )

andB A B AA B

B A B A

F p p F p pp c p cf p p f p p

− − −= + = +− −

. (4)

For example, the uniform distribution on [ ]HL, satisfies Assumption 1 and

generates 3/)2(* LHp A −= and 3/)2(* LHpB −= .

One final possibility is that BA qq ≤ and large customers satisfy both loyalty

programs. In that case, large customers pay a discounted price for every purchase,

and only small customers buy at the spot prices, so the same first order conditions

apply: the equilibrium spot prices are *Ap and *

Bp .

Before presenting our first results, we introduce some additional notation.

Define )},,({ BAA pqzCS to be the expected consumer surplus per purchase received

by large customers when they buy from firm A if and only if Aqx ≤ , and let

),( BA ppCS be the consumer surplus per purchase received by customers that

satisfy neither program and buy from firm A whenever .AB ppx −≤ We have

9

∫∫ −++−−=H

q B

q

L ABAAA

A

dxxfpxRdxxfzxRpqzCS )()2()()2()},,({ (5)

( , ) ( / 2 ) ( ) ( / 2 ) ( )B A

B A

p p H

A B A BL p pCS p p R x p f x dx R x p f x dx

−

−= − − + + −∫ ∫ (6)

Our first proposition establishes that in equilibrium, exactly one firm offers a

non-trivial loyalty discount that customers accept. This is shown in two steps. First,

we establish that if both firms offer loyalty discounts that customers satisfy, then at

least one of them is trivial, meaning that *ii pz = for some }.,{ BAi ∈ Second, we

show that whenever customers satisfy neither firm’s loyalty program, each firm can

offer a profit-improving loyalty program that customers will accept.

PROPOSITION 1: For the single market case,

(i) If customers satisfy both loyalty programs in equilibrium, then at least

one program offers no actual discount, that is *AAA ppz == or

*BBB ppz == .

(ii) In equilibrium, at least one firm offers a non-trivial loyalty program that

customers accept.

(iii) Therefore, exactly one firm offers a non-trivial loyalty program that

consumers accept in equilibrium.

PROOF: See Appendix.

Trivial loyalty programs in which firms offer no discount have no effect on the

market, and can be ignored. Thus Proposition 1.i effectively rules out equilibria in

which customers adopt both loyalty programs. Proposition 1.ii establishes that any

combination of pricing strategies in which customers choose to satisfy neither

program will always be undercut by one of the firms with a profit-improving

program that customers will adopt. Therefore, in equilibrium firms offer prices and

programs such that customers choose to satisfy (only) one loyalty program. We next

characterize the equilibrium, focusing in turn on the loyalty requirements ),,( BA qq

the spot prices ),,( BA pp and the loyalty discounts ).,( BA zz

10

PROPOSITION 2: Define },min{ cpHh B −= and },max{ ApcLl −= . In equilibrium,

hq A = and .lqB =

PROOF: We establish something slightly stronger. Namely, firm A’s best response to

any strategy of firm B always satisfies hq A = , and similarly for firm B. Proposition

1 implies that only one firm’s loyalty program is adopted in equilibrium. We first

establish the result for each firm when consumers satisfy its loyalty program in

equilibrium, and then turn to the case when the other firm’s program is adopted.

Suppose customers adopt firm A’s program in equilibrium, and let CS be a

large customer’s surplus under an initial profile of prices and loyalty programs. The

Lagrangian for firm A’s maximization problem is:

.)()()()(

)()()()(

21

21

−−++−−+

−−+−=

∫∫H

q B

q

L A

ABAAAA

A

A CSdxxfpxRdxxfzxR

ppFcpqFczL

λ

θ (7)

The first order conditions for Az and Aq are

0)()1( =−= AA

A

qFdzdL

λ (8)

[ ] 0)()( =−−+−= AAABAA

A

qfqzpczdqdL

λ (9)

The solutions to (8) are LqA = and 1=λ . We ignore the former solution

because it corresponds to a trivial loyalty program. Note that the latter solution

ensures that the consumer surplus from the loyalty program equals CS .

Substituting 1=λ into (9), and recalling that 0)( >xf for all ],[ HLx ∈ , establishes

that cpq BA −= solves (9). If Hcp B >− , then the objective is increasing in Aq so

Hq A = is optimal. Thus in equilibrium, hq A = when consumers adopt firm A’s

loyalty program. A similar calculation establishes that lqB = when consumers

satisfy firm B’s program.

11

Now consider the firm whose program is not adopted in equilibrium. The

alternative level of consumer surplus that firm A must provide to large customers in

order to have them participate in firm A’s program ( CS ) is obtained either from

adopting firm B’s loyalty program (large customers pay Ap and Bz ), or from

satisfying neither program (large customers pay Ap and Bp ). In equilibrium, the

alternative cannot be satisfying neither program, because if it were, firm B could

offer a profit-improving program that large customers would accept. That is, firm B

initially earns ))](1()(1[)( ABABB ppFqFcp −−+−−=Π′ θ and large customers

enjoy the consumer surplus as if they faced Ap and Bp . If all customers faced Ap

and Bp (loyalty programs not feasible), firm B would earn

))](1()(1[)( ABABBB ppFppFcp −−+−−−=Π θ which, using cpq BA −= and

,cp A > weakly exceeds BΠ′ . Proposition 1.ii implies that firm B can raise its profit

above BΠ (and hence above BΠ′ ) by offering a program that large customers will

adopt (because it provides at least the same consumer surplus as paying Ap and

Bp ). Thus in equilibrium, the CS constraint that firm A faces must be based on an

alternative in which large customers adopt firm B’s loyalty program. If customers

adopt firm A’s program in equilibrium, can firm B’s program be anything other than

a best response to firm A? No, because if B’s loyalty program is not a best response

in the candidate equilibrium, then firm B can reduce Bz so that large customers

adopt it instead of firm A’s program, and BΠ increases. Thus, in equilibrium, firm B

must offer a best response loyalty program, even if customers do not adopt it. From

above, this implies that .lqB = A similar argument establishes that hq A = (even)

when large customers adopt firm B’s program in equilibrium. £

Note that h and l determine whether any loyal purchasers always buy from

one firm. Specifically, h demarcates those purchases for which the customer prefers

12

firm B’s product at Bp even if firm A’s product is sold at marginal cost, while l

demarcates the analogous set of purchases that always go to firm A at Ap .12

As the proof of Proposition 2 establishes, the binding consumer surplus

constraint in an equilibrium in which large customers adopt firm A’s program is the

surplus they would obtain from firm B’s program. Using Proposition 2, the consumer

surplus from just satisfying firm B’s loyalty program is:

∫∫ −++−−=H

l B

l

L A dxxfzxRdxxfpxRCS )()()()( 21

21 (10)

Substituting this term into firm A’s maximization problem (7), differentiating with

respect to Ap , and substituting 1=λ yields

)()()()(0 lFppfcpppFdpdL

ABAABA

A

+−−−−== θθ (11)

In a candidate equilibrium in which consumers satisfy firm A’s loyalty program,

firm B sets Bp to maximize:

)](1[)()](1[)( ABBABB ppFcpqFcp −−−+−−=Π θ (12)

Differentiating with respect to Bp and substituting hq A = yields

[ ] 0)()()(1)(1 =−−−−−+−=Π

ABBABB

B ppfcpppFhFdpd

θ (13)

One can rewrite (11) and (13) in the following fashion:

12 Recall that consumers prefer A whenever AB ppx −< , and thus cpx B −> demarcates

the region in which consumers prefer B when A is priced at marginal cost. The definition of

h accounts for the possibility that cp B − may not lie in ],[ HL , or in words, that all

consumer types prefer B at Bp , even when .cp A = A similar explanation holds for .l

13

)()(1

)()(1

)()(

)()(

ABAB

ABB

ABAB

ABA

ppfhF

ppfppF

cp

ppflF

ppfppF

cp

−−

+−

−−+=

−+

−−

+=

θ

θ (14)

Since Ap c> and Proposition 2 together imply that A B Aq p p> − , the loyalty

contract is strictly binding. Our next proposition establishes that both firms price

above the Bertrand reaction function that arises when loyalty programs are not

feasible. Before presenting the proof, we explain the different reason each firm has

for abandoning the Bertrand reaction function. If large customers fulfill firm A’s

program in equilibrium, then firm B sets *BB pp > because it earns additional profits

from the remaining large customer purchases going to firm B. Since firm A’s loyalty

program is binding, all large customer purchases still made at firm B are for clients

that strictly prefer firm B’s product given prevailing prices. Thus, sufficiently small

increases in Bp above *Bp do not generate any lost large customer sales, and hence

are profitable for firm B. Turning to firm A, since large customers adopt firm A’s

loyalty program, no large customer sales are made at Ap . Nevertheless, a small

increase in Ap above *Ap relaxes the consumer surplus constraint on its loyalty

program price without having a first order effect on profits from small customers.13

When large customers adopt firm B’s loyalty program, firm B’s Lagrangian is

13 An analogy can be made to network competition. Satisfying a loyalty program is like

joining a network, buying a good at a loyalty program price is like making an on-net call, and

buying at a spot price is like making an off-net call. Raising the cost of access to network A

causes network B to charge its customers higher prices for off-net calls (that terminate on

network A). This reduces the surplus that network B provides to customers, so either

network B reduces other prices, or network A can profitably raise its prices. See, for example,

Laffont, Rey, and Tirole (1998).

14

−+−−−

−−++−−+

−−−+−−=

∫∫

∫∫H

q B

q

L A

H

q B

q

L A

ABBBBB

A

A

B

B

dxxfpxRdxxfzxR

dxxfzxRdxxfpxR

ppFcpqFczL

)()()()(

)()()()(

)](1)[()](1)[(

21

21

21

21λ

θ

(15)

and firm A profits are:

)()()()( ABABAA ppFcpqFcp −−+−=Π θ (16)

Differentiating these with respect to own spot price, substituting ,1=λ and using

Proposition 2, the corresponding first order conditions 0=Π AA dpd and

0=BB dpdL can be expressed as (14). Thus the equilibrium spot prices do not

depend on which loyalty program consumers adopt. Only the reasons each firm

raises its price are reversed. Since spot prices (weakly) increase with the

introduction of loyalty programs, small customers are harmed.

PROPOSITION 3: If HcpB =−* and ,* Lpc A =− then ),(),( **BABA pppp = in

equilibrium. Otherwise *AA pp > and .*

BB pp >

PROOF: When HcpB =−* and ,* Lpc A =− the system of equations (14) matches the

system of equations (3), and therefore the equilibrium spot prices when loyalty

programs are feasible match the spot prices when loyalty programs are not feasible,

i.e. ).,(),( **BABA pppp = Let )(⋅ir and )(ˆ ⋅ir respectively denote firm i ’s reaction

functions when loyalty programs are not feasible, and when they are feasible. Claim

).()(ˆ BABA prpr ≥ Suppose not, so there exists a p for which ).()(ˆ prpr AA < Using (3)

and (14), this implies that

.))(ˆ())(ˆ(

))(ˆ())(ˆ(

))(ˆ())(ˆ(

))(())((

prpfprpF

prpfprcF

prpfprpF

prpfprpF

A

A

A

A

A

A

A

A

−−

≥−

−+

−−

>−−

θ

15

Assumption 1 then implies that )(ˆ)( prpprp AA −>− which contradicts

).()(ˆ prpr AA < Therefore ).()(ˆ BABA prpr ≥ A similar calculation establishes that

).()(ˆ ABAB prpr ≥ These inequalities are strict when HcpB <−* or .* Lpc A >−

Let ),( BA pp be part of a candidate equilibrium when loyalty programs are

feasible, and assume that .*AA pp ≤ If ),(ˆ BAA prp = then ).( BAA prp > Claim that

)(*BAAA prpp >≥ implies that ).( ABB prp < Suppose not, so )( BAA prp > and

).( ABB prp ≥ If reaction curves intersect at ),,( **BA pp then for some ,*

AA ppp ≤≤ we

must have .)(

1])([)( 1

prprpr

AAB ′

=′>′ − The proof of Proposition 1, however, establishes

that 1)( <⋅′ir for ,, BAi = so we have a contradiction. Thus ),( BA pp cannot be an

equilibrium, or said differently, *AA pp > . Reversing the firm identifiers and

repeating the same argument establishes .*BB pp > £

Proposition 2 determines the equilibrium loyalty requirements—firm A requires

that customers obtain a )(hF share of their purchases from firm A, while firm B

requires customers to purchase a )(1 lF− share. Bertrand competition determines

the prices in these contracts. Suppose that, in the proposed equilibrium, customers

satisfy firm A’s program. If firm B can offer an alternative contract that guarantees

large customers the consumer surplus that they obtain from firm A’s loyalty

program while increasing its own overall level of profits, it will do so. The proof of

Proposition 2 shows that firm B’s optimal deviation generates the same large

customer surplus as under the proposed equilibrium. That determines firm B’s

discount to large customers when it deviates from the proposed equilibrium.14 In

equilibrium, firm A sets Az so that large customers enjoy the same surplus from

adopting firm A’s program as they would from switching to firm B’s program, and

firm B earns the same profit as it would from its optimal deviation. Note that at a

14 At the same time, firm B can change its spot price to obtain additional profits from small

customers, since large customers no longer buy at Bp . However, as noted above, the optimal

spot price for firm B does not depend on whose requirements contract large customers adopt.

16

fixed Bp , firm B would sell more if large customers abandoned both loyalty

programs because firm A’s program requirement )( Aq is strictly binding. Thus this

constraint ensures that at prevailing prices customers prefer firm A’s loyalty

program to no loyalty program at all. The equilibrium satisfies:

({ , }, ) ( ,{ , })A B A BCS z h p CS p z l= , and (17)

({ , , }, ) ( ,{ , , })B A A B B A B Bz h p p p z l pΠ = Π . (18)

Equations (17) and (18) determine the equilibrium values of Az and Bz . These can

be written as

[ ] [ ] ∫+−=−−−h

lAABB dxxfxlFphFzhFplFz )()()()(1)(1 (19)

[ ] [ ])(1)()(1)( lFczhFcp BB −−=−− , (20)

and solving them yields

( ) 1( ) ( )

( ) ( )

h

A A l

F lz c p c x f x dx

F h F h= + − − ∫ (21)

1 ( )( )

1 ( )B B

F hz c p c

F l−

= + −−

. (22)

Equations (14), (21) and (22) identify the equilibrium prices when large

customers adopt firm A’s program in equilibrium. If customers adopt firm B’s

program instead of firm A’s, the equilibrium spot prices ),( BA pp and loyalty

requirements ),( BA qq do not change. The equilibrium loyalty program prices are

( )( )

( )A A

F lz c p c

F h= + − , and (23)

1 ( ) 1( ) ( )

1 ( ) 1 ( )

h

B B l

F hz c p c x f x dx

F l F l−

= + − +− − ∫ . (24)

17

For “short-tailed” distributions, in which *Bp c H− ≥ or *

Ac p L− ≤ so that

1)( =hF or ( ) 0F l = ,15 no large customers buy at the spot prices in equilibrium. For

such distributions, the equilibrium contract prices, when firm A’s loyalty program is

adopted in equilibrium, are

,)()()( ∫−−+=h

lAA dxxfxlFcpcz and (21a)

.czB = (22a)

For short-tailed distributions, when firm A offers the chosen loyalty program, firm B

makes no profit from large customers. Thus it undercuts firm A’s program by setting

its discount price Bz at marginal cost. If, in addition, F is symmetric with mean

zero, then both firms set their loyalty discount prices at marginal cost, and neither

firm profits from large customers. If ( )F ⋅ is symmetric with a negative mean,

implying that customers on average favor firm A’s product, then only firm A’s loyalty

program is accepted in equilibrium. Conversely, when the mean of ( )F ⋅ is positive,

the only equilibrium has customers satisfying firm B’s program.

With “long-tailed” distributions, the firm whose loyalty program is rejected still

earns positive profit from large customers in equilibrium, so it does not set its

unfulfilled loyalty program price at marginal cost. This affords some pricing cushion

to the firm whose program is adopted. This implies that there are two distinct pure

strategy equilibria when the mean of the preference distribution is close to zero, and

consumers adopt each firm’s program in one of the equilibria. As the mean of the

distribution diverges from zero, the firm with the on average less preferred product

must make deeper and deeper price cuts to induce customers to adopt its program.

With sufficiently strong preferences for one product, the required price falls below

marginal cost, and the only equilibrium has large customers adopting the program

of the firm selling the preferred product. The next proposition establishes that

profits fall when loyalty programs become feasible for symmetric firms.

15 The uniform distribution, for example, is a short-tailed distribution.

18

PROPOSITION 4: When the distribution of reservation prices is symmetric with mean

zero, firms earn at least as much in equilibrium when loyalty programs are not

feasible as in the equilibrium when loyalty programs are feasible.

PROOF: When F is symmetric with mean zero and loyalty programs are infeasible,

the equilibrium prices from (3) become 1*** )]0(2[)0()0( −+=+=== fcfFcppp BA

and each firm earns .)]0(4[)1()0())(1( ** fFcp θθπ +=−+= When symmetric

firms can employ loyalty programs, they offer the same prices p and z , and earn

the same profits π , even though consumers satisfy only one program in equilibrium.

In the short-tailed case, *p p= and z c= , so firms earn the same profits as in the no

loyalty program equilibrium from small customers, while earning zero profits from

large customers. Thus *π π< . In the long-tailed case, (14) implies that

* *( )/( (0))p p F c p f pθ= + − > , while (21) and (22) imply that

( ) ( ) / ( )z c p c F c p F p c= + − − − . Profits are greater from small customers, but that

is countered by aggressive competition for large customers. Overall,

)()()( 21 cppcFcp −+−−= θπ . One interpretation is that this is firm A’s profits

when firm B’s program is adopted by large customers. For this case pcqB −= so

the first term is the profit from selling to large customers (who purchase a )(1 BqF−

share from firm B). The second term is the profit earned from selling to one half of

the small customers. Using the equilibrium values of p and ,*p this can be

rewritten as ))0((])([ 221 fpcF θθπ +−= . Thus *π π> only if

)]0(4[)1())0((])([ 221 ffpcF θθθ +>+− . Simplifying, this can be expressed as

24 ( ) 4 ( )F c p F c pθ θ− + − > . (25)

Suppose loyalty programs are not feasible, firm A charges ,* cpp −+ and firm B

charges .*p With these prices firm A makes sales whenever ,pcppx AB −=−≤

and thus earns ].2[)()1( * cpppcF −+−+ θ By definition of equilibrium, these

profits must be weakly less than ,)]0(4[)1( fθ+ firm A’s profits when it charges

19

.*p Using the definitions of p and *p above, this profit inequality can be written as

24 ( ) 4 ( )F c p F c pθ θ− + − ≤ , which contradicts (25). Thus *π π≤ . £

When F has mean sufficiently close to zero, firms generally earn less profit in

the loyalty program equilibrium than they do when neither firm can offer such

programs. When preferences for one brand are sufficiently strong, the preferred firm

does better using a loyalty program, and in some cases the less preferred firm also

benefits due to higher spot prices paid by small customers.

The consumer surplus implications are ambiguous. Spot prices increase in the

loyalty program equilibrium, so small customers are definitely harmed. While

loyalty programs reduce profits when firms are symmetric, large customer consumer

surplus does not necessarily increase due to the substantial number of purchases

shifted from what would be the preferred product in the absence of a share

requirement. Generally, when preferences for one brand are sufficiently strong,

allowing loyalty programs lowers overall consumer surplus.

3. Bundling across parallel markets

Suppose the market conditions described in Section 2 occur in a number of

parallel markets. That is, assume that customers make multiple purchases in

several related markets (such as different destinations from a hub airport) and a

firm can use a loyalty program that links purchases in all markets. Large customers

enjoy a discount only if they meet the share requirement in every market. A

customer that misses the target share in any one market loses the discount in all

markets. How does this potential to link across markets change the equilibrium?

It is helpful at first to make some simplifying assumptions (these can be

relaxed, as will be discussed shortly). Assume that there are N markets symmetric

in all characteristics, and that large customers buy products in all markets. A key

assumption is that firm A competes in all N markets, but that rival firm iB

20

competes just in market .i 16 Of firm A’s rivals, we assume that kBB ,...,1 )( Nk ≤

have the capability to offer loyalty discounts, and we study how equilibria depend on

.Nk We focus on equilibria in which customers adopt firm A’s program.

Let ,iAp ,i

Bp ,iAq ,i

Bq ,iAz ,i

Bz ,ih and il be the market i analogues to the

variables introduced in the previous section. The Lagrangian for firm A,

corresponding to (7), is

[ ]

−

−++−−

+−−+−=

∑ ∫∫

∑

=

=

CSdxxfpxRdxxfzxR

ppFcpqFczL

N

j

H

q

jB

q

L

jA

N

i

iA

iB

iA

iA

iA

A

jA

jA

121

21

1

)()()()(

)()()()(

λ

θ

(26)

The first order conditions 0=iA

A dzdL and 0=iA

A dqdL match (8) and (9), so as in

Proposition 2, 1=λ and .iiA hq = The argument in the second part of the proof of

Proposition 2 can be applied to each of ,,...,1 kBB so CS must be the surplus

available to large customers that satisfy all the loyalty programs of .,...,1 kBB Thus,

as in Proposition 2, iiB lq = for such rivals. The first order conditions 0=i

AA dpdL

and 0=Π iB

iB dpd match (11) and (13), so the spot prices are determined by (14),

and we can drop the superscripts on ,iAp ,i

Bp ,iAq ,i

Bq ,ih and .il In words, linking

parallel rivalrous markets together into a single loyalty program has no effect on

spot prices or loyalty requirements.

We now turn to the loyalty program prices. The procedure for finding iAz and

iBz is the same: firm A sets prices in a manner that prevents firm ,jB ,kj ≤ from

profitably offering a loyalty program that provides large consumers the same

consumer surplus as in the proposed equilibrium. Equation (18), which compares a

rival firm’s equilibrium profits to the profits from its optimal deviation, does not

16 For example, firm A may be a dominant airline at a hub airport while each rival serves

only a single spoke from the hub.

21

change. This implies that iBz remains the same (and we can drop the superscript).

The consumer surplus constraint (17), however, does change. If rival firms kBB ,...,1

collectively offer a sufficiently attractive alternative to the proposed equilibrium,

customers will abandon firm A in order to satisfy the program terms of kBB ,...,1 .

But by assumption, customers that do not meet firm A’s loyalty requirement in any

market lose the discount in all N markets. Thus the lower prices offered by

kBB ,...,1 must compensate not only for the lost discount in those k markets, but

also for the lost surplus in the remaining kN − markets. Condition (17) becomes

),()(}),{,()},,({1

BABA

N

iB

iA ppCSkNlzpCSkphzCS −+=∑

=

(27)

Letting ∑= iANA zz 1 and dividing through by ,N this constraint can be expressed as

),(1}),{,()},,({ BABABA ppCSNk

lzpCSNk

phzCS

−+= . (28)

Equation (28) shows that the average large consumer surplus from firm A’s loyalty

program is the weighted average of the consumer surplus from a rival program

(offered by ,jB kj ≤ ) and the lower consumer surplus from the higher spot prices in

markets that cannot have a rival program (markets Nk ...,,1+ ). Observe that (28)

matches (17) when .Nk = In such cases, equilibria of the multiple market setting

are simply N-fold copies of a corresponding equilibrium of the single market case. In

general, as N increases or k decreases, additional weight is placed on the last term

in (28) and firm A responds by raising its loyalty program prices. This is captured in

our next result.

PROPOSITION 5: As N increases or k decreases, the average loyalty program price Az

increases. This reduces large consumer surplus and overall consumer surplus.

PROOF: As noted above, Ap , Bp , Aq , ,Bq ,Bz ,h and l do not depend on k or .N

Let Nk

−= 1γ . To establish that increasing N or decreasing k reduces large

22

customer surplus, it suffices to show that ),(}),{,()1( BABA ppCSlzpCS γγ +− is a

declining function of γ , or that .0}),{,(),( <− lzpCSppCS BABA Using (6) and

∫∫ −++−−=H

lB

l

LABA dxxfzxRdxxfpxRlzpCS )()()()(}),{,( 2

121 , (29)

we have

0)](1[)()]()([)(

)(1)()(

)()()(

)(1)()()()]()()[(

)(1)()(

)()]()()[(

)](1[)()]()()[(

)](1[)()()(

}),{,(),(

≤−−−−≤

−

−−−−−=

−

−−−−−−≤

−

−−−−−−−=

−−−−−−−<

−−−−−=

−

∫−

lFlFlFhFcp

lFlFhF

lFppFcp

lFlFhFcplFppFcp

lFlFhF

cplFppFlpp

hFzplFppFlpp

hFzpdxxfxpp

lzpCSppCS

B

ABB

BABB

BABAB

BBABAB

BB

pp

lAB

BABA

AB

The first inequality above follows from the integrand being bounded above by

).()( xflpp AB −− The next equality uses (22). The first weak inequality uses

Apcl −≥ and ),()( lFppF AB ≥− and the last line uses ).()( AB ppFhF −≥ Thus

}),{,(),( lzpCSppCS BABA < , and increasing N or decreasing k diminishes large

customer surplus. Since small customer surplus is invariant, overall consumer

surplus must also decline. Given that h and Bp are invariant to k and ,N a fall in

)},,({ BA phzCS implies that Az must have increased. £

Although this discussion presumes symmetric markets, all that is really needed

is a lack of large customer heterogeneity. When elements like a large customer’s

preference distribution and purchase volume differ across markets, spot prices and

loyalty requirements will match the case when firms compete in each market

separately, though they may differ across markets. Adding a market that can offer a

23

loyalty discount tightens the consumer surplus constraint in (17) by increasing the

surplus available to large customers when they adopt the loyalty programs of firms

kBB ,...,1 instead of firm A. In contrast, adding a market that cannot offer a loyalty

discount relaxes the consumer surplus constraint in (17) by increasing the surplus

that customers forego when they abandon the dominant firm’s program. The latter

scenario allows the dominant firm to offer less attractive loyalty program discounts.

3.1 Exclusion: A First Look Proposition 1 establishes that (short-term) profit-maximizing firms employ

loyalty programs in equilibrium. While the overall surplus effects depend in part on

brand preferences, one cannot conclude that such pricing strategies are necessarily

anti-competitive. Dominant firms, however, may use loyalty programs with an eye

toward weakening or excluding rivals.

For a fixed mass of large customers, a firm can increase its share from current

loyal customers by raising the loyalty requirement. If the initial program has not left

money on the table, then the firm must offer an improved loyalty discount if it hopes

to retain customers while demanding increased loyalty. As long as the firm with the

adopted loyalty program sells the favored product, its discounted price exceeds cost

and there is room to cut price. That means that, at least for small changes, the new

price is not predatory using an (Areeda-Turner) price versus marginal cost test. If

the incremental discount is included in the incremental cost of serving the

additional customer, however, then it is possible that the firm would fail a price

versus incremental cost test. Specifically, for each additional purchase the firm

generates by raising the loyalty program target, it must compensate customers with

an incremental discount of A A Bz q p+ − , otherwise customers will switch to firm B.17

Including this incremental discount as a cost, the cost per purchase from an increase

in the program target )( BAA pqzc −++ exceeds the price )( Az whenever

17 Switching one purchase from B (at Bp ) to A (at Az ) reduces consumer surplus by

.21

21

BAAAABA pqzzqRpqR −+=++−−+ Since firm A’s equilibrium loyalty program

generates the same consumer surplus as firm B’s program, Firm A must compensate the

consumer by this amount to retain the customer as a loyalty program participant.

24

A Bq p c> − . This, according to Proposition 2, is the equilibrium share requirement

when firms employ loyalty contracts to maximize short-term profits apart from an

exclusionary motive. Therefore, this test is the proper tool to distinguish legitimate

from excessive loyalty program sales targets.

This incremental cost test, with incremental discounts counted as costs, is

essentially what Douglas Bernheim used as expert for the plaintiff in Virgin

Atlantic. Thus our analysis provides an analytical justification for a tool that has

already been introduced in antitrust litigation. The District Court in Virgin Atlantic,

as affirmed by the Second Circuit, ultimately rejected the implementation of

Bernheim’s test, but did not reject the test itself.18

Virgin Atlantic, of course, arises in a multiple market context. Our analysis

gives some sharp welfare results when loyalty contracts are extended from one

market to multiple markets. If rivals in the added markets cannot offer loyalty

programs, then in equilibrium customers receive less attractive loyalty discounts

and consumer surplus drops. From that perspective, there does not appear to be any

societal benefit from having loyalty contracts that link multiple markets. But the

anti-competitive effect of linked loyalty contracts is limited to this increase in

market power. Specifically, as the number of linked markets increases, there is

neither an increase in the dominant firm’s market shares nor any decrease in rival

profits or market share. Loyalty contracts are not any more exclusionary when

linked across multiple markets than when implemented in markets separately. Thus

the Bernheim test would be the appropriate standard in the multiple market context

as well: if a firm sets the target level so high that it loses money on incremental

18 As an expert for the plaintiff in Ortho Diagnostic Sys. v. Abbott Lab., F. Supp. 145 (1993),

Janusz Ordover advocated a similar test that includes rebates (or loyalty discounts) as an

incremental cost. See Ordover and Willig (1981) for an introduction to their concept of

“compensatory pricing”, and Greenlee, Reitman, and Sibley (2004) and Nalebuff (2004b) for

discussions of this price test applied to bundled rebates and several cases including Ortho.

25

sales, then there is a valid inference of exclusionary intent.19 Otherwise the use of

loyalty programs in single markets or multiple linked markets is consistent with

short-term profit maximizing behavior.

4. Bundling Monopolists

Since the dominant firm benefits from linking its loyalty program across several

markets, the next logical step is to consider whether linking a loyalty program to an

unrelated market is beneficial as well. In the previous model the linked markets

need not be for similar products or have common cost or production components, but

they had two common features: (1) common (large) customers who participate in

every market, and (2) the dominant firm faces a rival in each market. As long as

competition in each market induces the dominant firm to offer a loyalty program in

that market, it is (weakly) profitable to link the markets to common customers.

Suppose now that the dominant firm sells one or more products as a monopolist and

would not offer a loyalty discount unless it were linking together several products.

Does the monopolist profit from offering a loyalty program that links monopolized

and rivalrous markets?

Before characterizing loyalty programs that benefit the dominant firm, it is

worthwhile briefly to examine a bundled discount that does not. Consider a firm that

competes in a market like the one in Section 2, and is a monopolist in a second

market. Suppose this firm offers a rebate in the form of a lump sum deduction off

the bill for the monopolized product whenever the customer meets a loyalty target in

the rivalrous market. This rebate augments the consumer surplus that customers

can expect from meeting the loyalty program in (17). If each customer’s demand in

the rivalrous market is completely inelastic, then the other elements of the optimal

contract are unaffected, so the net result is that the loyalty program price in the

rivalrous market increases by an amount that exactly offsets the rebate. The fixed

deduction simply transfers money paid in the monopoly market to the rivalrous

market, with no effect on customer choices or effective prices. If market demand is 19 Investing in technology to track sales to more customers is another way to increase loyalty

program sales. Price/cost tests may not be useful to distinguish exclusionary from non-

exclusionary motives for such strategies.

26

not completely inelastic, such a transfer harms the monopolist because customers’

reduced purchases generate deadweight losses that cannot be recovered as profit.

The conclusion is quite different, however, when the monopolist offers customers

a per unit discount on purchases of the monopoly good. In order to focus on this role

for loyalty discounts, it helps to change several features of the model. We assume the

rivalrous market has limited or no perceived product differentiation, so that firms

would not profit from using a loyalty program apart from linking this product to the

monopolized market. We assume now that the market demand curve is downward-

sloping so that the disadvantage to bundling discussed above comes into play.

Finally, when two of its products are substitutes, the dominant firm may have an

additional incentive to link sales in order to rationalize the allocation of purchases

across products.20 To eliminate this potential advantage from bundling, we assume

the products are in separate product markets.21 Nevertheless, in keeping with recent

case history, we refer to the monopolized product as the branded product, and the

rivalrous market as the generic market.

The monopolist can try to limit competition in the generic market by linking

generic sales to branded sales. Specifically, the monopolist can offer a program that

provides a discount on branded purchases to customers that satisfy a loyalty

requirement in the generic market. The results of Section 2 suggest that when there

is limited product differentiation, the equilibrium loyalty program entails supplying

the entire market. Thus, we start by assuming that the monopolist makes an all-or-

nothing offer in the generic market, and later discuss what happens when the

monopolist only requires customers to buy a predetermined share of their generic

purchases from the dominant firm. The question is whether, or in what

circumstances, the additional generic profits exceed the branded market discount.

20 In several of the legal cases that address loyalty programs that link monopolized and

rivalrous products, the rivalrous product is a substitute for at least one of the monopolized

products. For example, transparent tape in LePage’s and cephalosporin antibiotics in

SmithKline. SmithKline Corp. v. Eli Lilly & Co., 575 F.2d 1056 (3rd Cir. 1978).

21 In LePage’s, for example, 3M linked private label tape sales to discounts offered across a

number of different product categories, including healthcare and retail automotive products.

27

Assume that demand in the branded market has two segments: large customers

have demand )( pL , also buy the generic product, and are candidates for the loyalty

program. Small customers, with demand )( pS , either do not buy generic products or

are not candidates for a program. Generic market demand is given by ),(kG and is

supplied by the monopolist and one or more competitors.22 Define )(yIε to be the

price elasticity of demand in market (segment) I at price y , and assume that each

of these demand curves has the usual property that 0)( <⋅′Iε , i.e. the point elasticity

of demand has greater magnitude at higher prices. Demand in each market segment

is comprised of customers who each buy multiple units. To simplify the analysis,

assume that the demand profile of individual customers across their various

purchases is similar, so that aggregate demand can be treated as if it is supplied to a

representative consumer. The monopolist produces branded products at marginal

cost ,c and all firms produce the generic product at marginal cost gc .

The monopolist’s profits in the generic market depend on such things as the

number of competitors in that market, the nature of competition, and the perceived

degree of product similarity. The monopolist can alter competition in the generic

market by offering to supply all customer requirements at a discounted price

without linking to the branded market. Like an auction for the entire market,

competing all-or-nothing offers effectively transform the market into pure Bertrand

competition. If customers do not prefer one supplier’s generic product to another,

then the equilibrium all-or-nothing price in the generic market is gc . Absent scale

economies, all-or-nothing offers for an undifferentiated product are not profitable.

Now suppose that the monopolist links sales in the branded market to the all-or-

nothing offer in the generic market. Customers who accept the offer get a discount

d off the regular price p on all branded market purchases, provided they agree to

buy generic product exclusively from the monopolist at price k . In order to insure

22 The model focuses on the generic product demand of customers who also buy the branded

product. There may be additional sales to generic-only customers that are not included in the

model and are not affected by loyalty discounts offered to customers who buy both products.

28

that customers buy generic product from the monopolist, customers must enjoy at

least as much combined consumer surplus in the branded and generic markets as

they would receive from buying branded products at the non-discounted price and

obtaining generic product from a rival firm at price gc . That is,

,)()()()( ∫∫∫∫∞∞

−

∞∞+≤+

kdpcpdxxGdxxLdxxGdxxL

g

or

∫∫ −≤

p

dp

k

cdxxLdxxG

g

)()( . (30)

The monopolist solves

)()()()()()(),,(max,,

kGckdpLcdppScpkdp gMkdp

−+−−−+−=Π (31)

subject to (30). The necessary conditions for a solution are

0)()()()()1()()()( =+−′−−+−−+′−+ pLdpLcdpdpLpScppS λλ (32)

0)()()()1( ≤−′−−−−−− dpLcdpdpLλ (33)

0)()()()1( =′−+− kGckkG gλ (34)

along with (30) and two complementary slackness conditions:

0)()( =

− ∫∫ −

p

dp

k

cdxxLdxxG

g

λ , and (35)

{ } 0)()()()1( =−′−−+−− dpLcdpdpLd λ , (36)

where 0≥λ is a Lagrange multiplier.

PROPOSITION 6: In equilibrium, the monopolist offers an all-or-nothing program that

includes a positive discount. Namely, the solution to (31) has 0>d .

29

PROOF: The monopolist never sets 0<d , since customers will ignore the “discount”

and buy at the posted price, p . Suppose 0=d . From (30), gck = , and so 1=λ from

(34). Substituting 1=λ into (32) implies that cp > . But then the left hand side of

(33) is positive, which is a contradiction. Therefore 0>d . £

Intuitively, a small discount below the monopoly price causes only a second

order loss in the monopolist’s profits from branded sales to large customers, but a

first order gain in profits in the generic market, and so is always advantageous.23

Since 0>d , (36) requires that (33) holds with equality. Therefore, substituting (33)

into (32) yields

0)()()()( =+′−+ pLpScppS λ . (37)

Letting )(yIµ be the markup in market (segment) I at price y , (33) can be written

as

)()()(

)()(1 dpdp

dpLdpLcdp

LL −−=−

−′−−=− εµλ , (38)

while (34) can be expressed as

)()()(

)()(1 kk

kGkGck

GGg εµλ =

′−=− . (39)

Combining (38) and (39),

)()()()( kkdpdP GGLL εµεµ =−− . (40)

The monopolist’s optimal all-or-nothing discount program is characterized by (30),

(35), (37), (40), and either (38) or (39).

According to Proposition 6, the monopolist always offers large customers a

discount off the branded price charged to small customers. What happens to small

customers when the monopolist introduces a loyalty program? Analogous to

23 Burstein (1960) and Mathewson and Winter (1997) establish a similar result for tying.

30

Proposition 3, the next result establishes that the monopolist increases the spot

price in order to accommodate the discount. Let op denote the equilibrium monopoly

price in the branded market when there is no discount program.

PROPOSITION 7: The optimal branded spot price when using an all-or-nothing

discount program in the generic market has opp > .

PROOF: The branded price with no discount program is determined by the first order

condition

0)()()()()()( =′−++′−+ pLcppLpScppS ooooo . (41)

Substituting (38) into (37), the first order condition for p is

0)()()()()()()( =−−++′−+ dpdPpLpLpScppS LL εµ . (42)

Evaluating the left hand side of (42) at op and using (41) gives

( ))()()()()( oL

oL

oL

oL

o ppdpdppL εµεµ −−− . (43)

Now by assumption 0)()( <−< dpp oL

oL εε , and 0)()( >−> dpp o

Lo

L µµ , so (43) is

positive which means that 0/ >Π dpd M at opp = . Thus opp > at the optimum. £

The discount linkage between the branded market and the generic market spills

over to small customers even though they only buy branded products. The linkage

unambiguously harms small customers because they pay higher prices. Large

customers get a discount off the branded price, and may appear to benefit, but that

benefit is illusory. Paying more in the generic market offsets the branded market

discount. At the optimum, condition (30) typically holds with equality. This ensures

that large customer surplus matches what they would obtain from paying the

undiscounted price in the branded market and buying the generic product from a

rival producer. Thus large customers, like small customers, are harmed because

opp > . Their net benefit relative to the equilibrium with no discount program

31

depends on the competitiveness of the generic market. If gck = , then large

customers are unambiguously harmed.

The preceding discussion asserted that (30) typically holds with equality. The

next proposition identifies conditions under which that is true, and further

characterizes optimal prices. To do this, it is helpful to define monopoly prices when

the monopolist can set prices to each segment separately. Let oSp , o

Lp , and ok denote

respectively the monopoly prices in the small branded, large branded, and generic

market segments, and define ∫∫ −=Γo

goL

k

c

y

pdxxGdxxLy .)()()(

PROPOSITION 8: The equilibrium all-or-nothing discount program prices satisfy:

(i) If ooS pp ≤ , then o

Spp > , oLpdp <− , and okk < .

(ii) If ooS pp > and 0)( <Γ o

Sp , then oSpp > , o

Lpdp <− , and okk < .

(iii) If ooS pp > and 0)( ≥Γ o

Sp , then oSpp = , o

Lpdp =− , and okk = .

PROOF: (i) By assumption ,ooS pp ≤ and so, using Proposition 7, o

Spp > . Since oSp

maximizes profits in the S market segment and oSpp > , it follows that

0)()()( <′−+ pScppS . Therefore, from (37), 0>λ , which implies, using (38), that

1)()( −>−− dpdP LL εµ . Since )(⋅Lµ is increasing, )(⋅Lε is negative and

decreasing, and 1)()( −=oLL

oLL pp εµ , it follows that o

Lpdp <− . Similarly, using

(39), 1)()( −>kk GG εµ , and thus .okk < (iii) If 0=λ , then (37), (38), and (39)

imply, respectively, that oSpp = , o

Lpdp =− , and okk = , and (40) is satisfied. If

0)( ≥Γ oSp , then at these prices, (30) and (35) are satisfied as well. Thus all of the

conditions for an optimum are met. (ii) Suppose, as in (iii), that 0=λ , which implies

that oSpp = , o

Lpdp =− , and okk = . But then 0)( <Γ oSp means that (30) is violated

at those prices. Thus we must have 0>λ , which implies that oLpdp <− and okk < .

Moreover, from (37), 0)()()( <′−+ pScppS , which implies that oSpp > . £

32

According to this proposition, the only circumstance in which (30) does not bind

occurs when the monopolist charges the monopoly price in each product segment. In

that situation, customers taken together are clearly harmed by the introduction of

an all-or-nothing loyalty program. Otherwise condition (30) holds with equality.

Proposition 8 highlights an additional motivation for the monopolist to link the

branded and generic markets with a discount program, which is to price

discriminate among branded customers. By giving large customers a discount, the

monopolist effectively introduces separate prices in the two branded market

segments. If small customers have more elastic demand than large customers, as in

Proposition 8.i, then the discount works against the optimal price discrimination

scheme. Nevertheless, the monopolist still profits from charging a lower price to

large customers than to small customers. Recalling that large customers are defined

as those who can be identified as buying the generic product, it is arguably more

plausible that small customers have less elastic demand than large customers.

Generic buyers may be inherently more price sensitive for the product overall, and

may also have more elastic demand for the branded product because they have

access to the generic alternative. If small customers have more inelastic demand,

then the discount program not only allows the monopolist to extract profits from the

generic market, but also to gain additional profits by price discriminating among

branded customers. A loyalty program can also provide a means to distinguish the

various demand segmentscustomers who enroll in the program are generic buyers

and are likely to have more elastic demand.

The next two results examine how prices change as the relative sizes of the

large and small customer segments of the branded market change. First suppose

that relatively few branded customers also buy the generic good. Specifically, define

}0)(|{ >= ySyPS and [ ].)()(sup ySyLSPy∈

=α . With this, we have the following result.

PROPOSITION 9: In the limit as 0→α , opp = .

PROOF: Equation (37) can be rewritten as

33

λαλεµ ≤=−−≤)()(

)()(10pSpL

pp S . (44)

As 0→α , (44) implies that 1)()( −→pp Sεµ , which in turn implies that oSpp → .

Also, as 0→α , (41) implies that oS

o pp → . Thus opp → . £

The loyalty program has a negligible effect on the branded price when few

branded customers also buy the generic good. The monopolist, however, still offers a

branded discount to generic customers, the size of which depends on the relative

demand for the two products among the customers that buy both.

Now consider the opposite extreme, in which almost all branded customers also

buy the generic product, and define },0)(|{ >= yLyPL [ ],)()(sup yLySLPy∈

=β and

}0)(|{infy

== ySyp S .

PROPOSITION 10: In the limit as 0→β ,

(i) If 0)( <Γ Sp , then Spp → , oLpdp <− , and okk < .

(ii) If 0)( ≥Γ Sp and 0)( <Γ oSp , then *pp → where 0)( * =Γ p ,

oLpdp →− , and okk → .

(iii) If 0)( ≥Γ oSp , then Spp → , o

Lpdp =− , and okk = .

PROOF: Now equation (37) can be rewritten as

[ ] 0)()(1)()(

=++ λεµ pppLpS

SS . (45)

(i) Note that (39), the non-negativity of λ , and the increasing elasticity assumption

together imply that okk ≤ . Since in this case 0)( <Γ Sp , (30) requires that

oLpdp <− , which from (38) means that 0>λ . Therefore, as 0→β , (45) implies

that −∞→)( pSε , which in turn implies that Spp → . Also, as in Proposition 8.i,

34

when 0>λ , okk < . (ii) With 0)( =Γ Sp , then in the limit with 0→λ , Spp → ,

oLpdp →− , and okk → , conditions (30) and (45), along with all the other

conditions for an optimum, are satisfied. Similarly, if 0)( >Γ Sp , then when *pp →

all the conditions for optimality are satisfied in the limit. (iii) First note that oS

o pp → as 0→β . With 0)( ≥Γ oSp , all the conditions for an optimum are satisfied

with 0→β as in proposition 9.iii. £

As small customers become less significant, the monopolist raises the spot price

in order to make the discount offered to large customers more compelling. In the

limiting case, the monopolist uses all the consumer surplus in the branded market to

leverage a higher price in the generic market.24

The results thus far have addressed the optimal prices in an all-or-nothing

discount program. The monopolist’s decision to use such a program, however,

depends on the price and share it would receive in the generic market when the

products are sold separately. Based on the preceding results, we can draw some

general conclusions. First, if the generic market were competitive, so that the

market price gk c= in the absence of a loyalty program, then the monopolist always

benefits from offering an all-or-nothing discount program. In fact, even if the rival

firms have a slight cost advantage over the monopolist, the monopolist still finds it

profitable to use a loyalty program, and its more efficient rivals are excluded.

Second, for any k , if the monopolist’s share at that price is sufficiently low, then the

monopolist also profits from introducing a discount program. In some situations

when the monopolist would like to price discriminate among branded customers

based on their use of the generic product, the monopolist can obtain monopoly profits

in both the branded and generic markets by using a loyalty discount program.

Finally, the monopolist tends to benefit from using loyalty discounts if (1) branded

customers who do not buy the generic product have more inelastic demand for the

branded product than customers who buy both generic and branded products, and

24 For further discussion, see Greenlee, Reitman, and Sibley (2004), and Nalebuff (2004a).

35

(2) either there are relatively few exclusively branded customers, or customers who

buy both products have larger demand for the branded product.

There are two distinct components to the monopolist’s profit gain from using a

loyalty program to link a rivalrous market to its monopolized market. First, the

monopolist can introduce a gap between the monopolized product price for program

participants and non-participants in order to extract additional profits from

customers who buy both products. Second, the monopolist can offer a discount on the

bundled product that is worth more to customers than it costs the monopolist

(because of increased sales), and thereby extract additional profits in the linked

market. Both components rely on the same mathematical property: the optimal

branded product price without loyalty discounts satisfies a first order condition, so

small changes in the price have no first order effect on profits, but do have a first

order effect on consumer surplus. One can construct variations of this model in

which only one of the effects is present. Thus even if customers do not make repeat

purchases, the monopolist still profits from introducing a discount program (in

effect, tying) that introduces a wedge between the participant price and the non-

participant price. Alternatively, even if customers eligible for the loyalty program

are of negligible size relative to the total market for the branded product so that, as

shown in Proposition 9, the spot price for the branded product does not change, the

monopolist still benefits from using a loyalty program.

The model implicitly assumes that the only alternatives available to the

dominant firm are to use a loyalty discount that links the branded market to the

entire generic market, or else to operate completely separately in the two markets

and charge a single price in the branded market. In equilibrium, the dominant firm

establishes a loyalty program in large part to extract additional profits in the

branded market. Alternative pricing strategies, however, may be available for

similar rent extraction. As discussed above, one advantage of the loyalty program is

that it operates as a segmenting device that enables price discrimination across

branded customers. When exclusively branded customers have more inelastic

demand, this tool increases profits. However, if alternative means to distinguish

customers exist, then the gains from using a loyalty program are diminished.

36

When customers make multiple purchases, the monopolist can use a two-part

tariff in the branded market. In the extreme case of no uncertainty or heterogeneity

in demand, or if a program can be tailored to each customer, the monopolist prices at

marginal cost and extracts all consumer surplus via a fixed fee.25 In such cases,

there is no incentive to link a loyalty program to the generic market because any

profit gained in the generic market would be sacrificed in the branded market, and

the monopolist could not recover the deadweight loss created in the generic market.

Moreover, the monopolist cannot use the loyalty program to threaten to charge a

higher price in the branded market, because the monopolist already extracts all

surplus—at a higher price, the customer makes no branded purchases.

Alternatively, suppose that the monopolist cannot extract all surplus from

consumers through a two-part tariff. Specifically, assume that there is demand

uncertainty or unobservable heterogeneity among customers so that the monopolist

maximizes an average across customer types. The familiar “Mickey Mouse

Monopolist” result (Oi 1971) is that, with some restrictions on demand across types,

the monopolist charges a per unit price above marginal cost along with a fixed fee.

In this setting, our analysis goes through unaltered. Namely, the monopolist still

has an incentive to offer a discount to branded customers that make all of their

generic purchases from the monopolist. The only difference is that the discount

applies to a price that satisfies the first order condition for the monopolist’s optimal

two-part tariff, rather than the monopoly price.

The model assumes that the monopolist can still offer a branded good discount

that restores the customer surplus lost through a higher generic price. This appears

to contradict the assumption of uncertainty or unobservable heterogeneity.

Implicitly, this interpretation of the model requires the demands in the branded and

generic markets to be correlated. As long as sales for the two products move

together, the monopolist does not need to know the scale of demand in order to

introduce a loyalty program that equates changes in consumer surplus across the

branded and generic markets. This suggests that the loyalty program will not be

25 Incidentally, the fixed fee could be implemented through a discount program similar to a

requirements contract, but in which the discount does not apply to every unit purchased.

37

implemented across products whose demand is completely independent. Even in a

case like LePage’s, in which transparent tape sales were linked to rebates on

categories like automotive and healthcare products, it is plausible that the demands

were correlated in a dimension unobservable to 3M because the purchases were

made at the same stores. That “same store correlation” could provide sufficient

incentive to implement at least modest discounts. It also suggests that linking sales

across multiple product categories, in order to better distinguish store-specific from

product-specific demand shocks, may be profitable for the firm.

4.1 Exclusion Revisited

In equilibrium, rival firms are completely excluded from generic product sales,

at least to branded product customers. This equilibrium, however, can be

implemented by requiring only a partial share of generic purchases and adjusting

the price accordingly. For example, if the equilibrium price k is one dollar over

marginal cost, the monopolist could require that large customers make half of their

generic purchases from the monopolist at two dollars over marginal cost. The

customer’s marginal purchase decision is the same, since for every two additional

generic units purchased, one must be from the monopolist. In this way, the

equilibrium can be implemented with any fixed share requirement, and both the

monopolist and the customer are as well off. This holds as long as rival firms do not

make profitable sales. If rival firms can price above cost when the monopolist uses a

share requirement below 100 percent, then the monopolist benefits from excluding

them completely. Alternatively, if rival firms have low marginal costs and would sell

at a price below the monopolist’s cost, then the monopolist has an incentive to give

rival firms a share of the market. Doing so lowers customers’ average purchase price

for generic products and increases the monopolist’s profits.

Instead of using a share requirement, the monopolist could instead require a

given volume of generic purchases to qualify for the discount. If rival firms sell the

residual at (or close to) marginal cost, then the monopolist actually prefers a volume

requirement because it reduces the deadweight loss in the generic market, and

increases the consumer surplus that can be captured through the linked volume

discount. Of course, if the monopolist can use volume targets, implying that there is

38

little uncertainty about customer types, then it may be able to extract profits using

alternative pricing methods, as discussed earlier.

Since complete exclusion is a natural outcome of this model, it is not clear what

it means to talk about too much exclusion. Nevertheless, one can still apply the

incremental cost test discussed earlier, particularly if the increment can be defined

relative to a time period before the monopolist linked the markets together in a

loyalty program. The result depends on the implementation of the test. One could

look at the incremental generic sales to customers who buy both products, where one

of the costs is the branded discount offered to these customers. Letting G denote the

monopolist’s equilibrium output prior to implementing a loyalty program, one could

compare ( ( ) )( )gG k G k c− − to ( )d L p d− . Observe, however, that

( ( ) )( ) ( ) ( ) ( )g

k p

g c p dG k G k c G x dx L x dx d L p d

−− − ≤ ≤ ≤ −∫ ∫ (46)

where the middle inequality is from (30). Thus the monopolist would always fail this

implementation of the test. If additional branded sales, to large customers or to both

large and small customers, are included in incremental sales then the inequality can

be reversed. It is not clear, however, whether any such calculation indicates that the

loyalty discount has crossed over from legitimate competition to unlawful exclusion.

The incremental sales test, however, can be used as a safe harbor. That is,

suppose that the monopolist’s pricing causes the variable profits it earns on generic

sales to exceed the total discounts for branded purchases. The next proposition

establishes that such prices could not exclude an equally efficient generic rival.26

Since we prove the result under slightly more general conditions (e.g. the branded

and generic products can be substitutes) and introduce a fair amount of additional

notation, the proof appears in the Appendix.

26 For additional discussion of price tests for bundled rebates, including ones based on

consumer surplus rather than excluding equally efficient competitors, consult Greenlee,

Reitman, and Sibley (2004), and Nalebuff (2004a, 2004b).

39

PROPOSITION 11: If the monopolist offers a loyalty program in which its revenues

from generic sales exceed the variable cost of generic sales plus the discount offered

on branded sales, then there exists a rival price for generic goods r, with ,gcr > such

that large customers would forego the loyalty discount and make all generic

purchases from a rival charging r.

PROOF: See Appendix.

The model necessarily omits other factors that affect whether it is efficient for

customers to sole source. For example, scale economies in manufacturing may make

dual sourcing undesirable. In addition, costs or network benefits on the customer

side may favor buying from a single supplier. Alternatively, customers may have

diverse preferences across the product offerings of several competitors. As the model

of Section 2 demonstrates, however, such diverse preferences do not prevent firms

from offering a loyalty program that covers all but the tails of the distribution.

5. Conclusions

Once loyalty programs are added to the arsenal of competitive tools available to

firms in a differentiated-product, price-competitive market, they take on a

prominent role. In a single market, loyalty discounts typically displace any

equilibrium in which firms compete just by setting spot prices. Multi-product firms

from whom customers buy products in several parallel markets have an incentive to

offer loyalty discounts that link purchases across markets. And finally, a monopolist

in one market that competes in a second market with common customers may have

an incentive to offer a loyalty discount across both markets in order (1) to squeeze

additional profits out of the monopolized market, and (2) to gain additional sales in

the rivalrous market. In each case, the equilibrium is characterized by aggressive

pricing to loyalty program participants, higher prices to other customers, and

significant share shifts to the firm whose loyalty program is adopted in equilibrium.

In the single market and parallel market cases, a simple incremental cost test can

distinguish competitive from intentionally exclusionary behavior.

40

6. Appendix

Proof of Proposition 1.

(i) If large customers satisfy both programs, then the equilibrium spot prices satisfy

(4), that is *AA pp = and *

BB pp = . First, we must have *AA pz ≤ and *

BB pz ≤ , or else

customers reject the loyalty program and purchase at the spot price. In addition, if

both loyalty programs can be satisfied simultaneously, then we must have BA qq ≤

and large customers pay a discounted price for every purchase.

Let )(⋅ir denote firm i ’s reaction curve in the Bertrand game without loyalty

programs. We first establish that 1)(0 <⋅′< ir for }.,{ BAi ∈ Let )()()( xfxFxH A =

and )())(1()( xfxFxHB −= . Using (3), firm A’s reaction function satisfies

))(()( BABABA prpHcpr −+= . Differentiating with respect to Bp and rearranging

yields .))((1

))(()(

BABA

BABABA prpH

prpHpr

−′+−′

=′ Assumption 1 states that 0)( >⋅′AH and this

implies that .1)(0 <⋅′< Ar Similarly, firm B’s reaction function satisfies

))(()( AABBAB pprHcpr −+= . Differentiating with respect to Ap and rearranging

yields .))((1))((

)(AABB

AABBAB pprH

pprHpr

−′−−′−

=′ Assumption 1 states that 0)( <⋅′BH and this

implies that .1)(0 <⋅′< Br

Moving on, 1)( <⋅′ir implies that ,)()( **BBBABA zpzrpr −<− or rewriting,

.)( **ABBAB ppzrz −<− Similarly, 1)( <⋅′Br implies that .)( **

ABAAB ppzzr −>− We

consider in turn the three possible orderings of ,Aq ,Bq and .**AB pp − Assume first

that .**BABA qppq <−< If ,**

ABAB ppzz −<− then firm B has an incentive to raise

Bz because its best response satisfies .)( **ABAAB ppzzr −>− Similarly, if

,**ABAB ppzz −>− then firm A has an incentive to raise Az because its best response

satisfies .)( **ABBAB ppzrz −<− Thus the only possible equilibrium satisfying

* *A A B Bq p p q< − < has .**

ABAB ppzz −=− Given that the slopes of both reaction

functions are less than one, the unique solution is *AA pz = and *

BB pz = .

41

Summarizing, if BABA qppq <−< ** in equilibrium, then neither firm offers an actual

discount. Now assume that **ABA ppq −≥ and consider in turn the possible orderings

of Aq and .AB zz − If ,AAB qzz >− then **ABAB ppzz −>− and firm A has an

incentive to raise Az because its best response satisfies .)( **ABBAB ppzrz −<− If

,AAB qzz =− then all purchases made at firm A are for customers that strictly

prefer firm A’s product at Az rather than firm B’s product at .Bz Thus firm A can

profitably increase Az without causing large customers to abandon the program. If

,AAB qzz <− then firm A’s loyalty program is strictly binding. Sales made by firm B

are to customers that strictly prefer B to A at prevailing prices ),( BA zz , so firm B

can profitably increase Bz without losing any customers. This establishes that

whenever **ABA ppq −≥ , one firm always has an incentive to increase its loyalty

discount price .iz Thus for this case we must have *AA pz = or .*

BB pz = From above,

1)( <⋅′Ar implies **ABAB ppzz −<− . Together with ,*

AA pz ≤ this implies that *BB pz <

and therefore .*AA pz = Summarizing, an equilibrium with both loyalty programs

satisfied and **ABA ppq −≥ can only exist if *

AA pz = . Finally, for ,**ABB ppq −≤ a

similar argument establishes that an equilibrium with both programs satisfied

exists only if .*BB pz =

(ii) Mathematically, we show that whenever neither loyalty program is adopted,

firm A has a profitable deviation that includes an accepted loyalty program. (A

similar calculation establishes a parallel result for firm B.) Suppose that large

customers allocate purchases with qx ≤ to firm A at price Ay and buy from firm B

at By when AB yyqx −=> with ,HqL << ,cy A > and .cyB > We show that

there exists a loyalty program },ˆ{ AA qz with },{min cyHqq BA −≤< such that (a)

),()},,ˆ({ BABAA yyCSyqzCS = , and (b) ˆ({ , }, ) ( , )A A A B A A Bz q y y yΠ > Π . Condition (a)

can be written as:

( /2) ( ) ( /2) ( )

ˆ( /2) ( ) ( /2) ( ) , orA

A

q H

A BL q

q H

A BL q

R y x f x dx R y x f x dx

R z x f x dx R y x f x dx

− − + − + =

− − + − +

∫ ∫

∫ ∫

42

ˆ( ) ( ) ( ) ( ) ( )Aq

A B A A B qz y F q y y F q x f x dx− = − − ∫ . (A1)

Let Az satisfy (A1). Now, writing out the expressions for firm A profits,

)()()()(),( and

)()()()()()()(

)()ˆ()()()},,,ˆ({

qFcyppFcpyy

dxxfxqFyyqFcyppFcp

qFczppFcpyyqz

AABABAA

q

qBAABABA

AAABABAAAA

A

−+−−=Π

−−+−+−−=

−+−−=Π

∫θ

θ

θ

Taking the difference,

,0)()(

)()]()([)(),()},,,ˆ({

>−−=

−−−=Π−Π

∫

∫A

A

q

q B

q

qABBAABAAAA

dxxfxcy

dxxfxqFqFcyyyyyqz

since the integrand is positive between q and Aq . Thus },ˆ{ AA qz also satisfies (b).

(iii) Follows directly from (i) and (ii). £

Proof of Proposition 11.

We generalize the demand characterization from the model in the text by

assuming that each customer has a multitude of potential purchases characterized

by ,b the value of buying the branded good for that purchase, and ,g the value of

buying the generic product for that purchase. Let ),( gbF describe the joint

distribution over the consumer’s potential purchases. For simplicity, the generic

products from the monopolist and from rivals are regarded as perfect substitutes,

although this is not critical for the result. The customer allocates potential

purchases to the branded product, to the generic product, or to no product. As before,

let p denote the branded product spot price, d the loyalty discount on branded sales

for customers that purchase the target level of generic product from the monopolist

at price ,k and gc the monopolist’s marginal cost for the generic product.

Suppose the customer switches from participating in the loyalty program to

buying branded products at the spot price and buying all of its generic requirements

43

from a rival at price .gcr > We can partition the set of potential purchases based on

the allocation made under the two pricing schemes (“before” refers to purchases

made under the loyalty program, “after” refers to purchases made at spot prices):

N = the number of potential purchases for which the customer bought neither

product before, and then bought the generic product from the rival afterward.

G = generic purchases made from the monopolist before and from the rival after.

R = generic purchases made from the rival before and after.

S = branded purchases before, and generic purchases after.

B = branded purchases both before and after.

X = branded purchases before, and neither branded or generic after.

Z = neither branded or generic purchase before or after.

We need two additional technical assumptions. First, there are potential

purchases in each of these partitions. Second, when the customer gets the same

surplus from either buying or not buying the product at given prices, we assume

that the customer either always prefers to buy or else always prefers not to buy.27

Given this partition, the safe harbor rule can be characterized as

).()( XBSdGck g ++≥− (A2)

Note that the safe harbor is an ex post test. Namely, whether or not it is satisfied

depends on the number of purchases made of each product. The main result here is

that, if the safe harbor is satisfied, then a rival generic firm that prices sufficiently

27 Making the lexicographic preference assumption avoids having all customers indifferent

between two options, so that the surplus gain is zero even when the rival prices at marginal

cost. Alternatively, one can assume that some customers are not on a boundary between

partition regions, which would hold if the density function is positive over a range of

reservation values.

44

close to the monopolist’s marginal cost can always induce customers to reject the

monopolist’s loyalty program.

The customer will abandon the loyalty program if buying under the spot

prices yields a higher surplus. It is sufficient to look at the surplus change for each

of the purchase groups identified above.

N – The average gain in surplus is ,v with ,0 rkv −≤≤ so that the total gain in

surplus is .Nv

G – The gain is rk − per purchase.

R – There is no change in surplus.

S – The surplus before was ,dpb +− and after is rg − with .pbrg −≥− Therefore

the average change in surplus per purchase is ,ddpbrg −=−+−− σ with

.0≥σ

B – The change in surplus is d− per purchase.

X – The change in surplus is ,d−µ with .0 d≤≤ µ

Z – There is no change in surplus.

Thus the total change in surplus from dropping the loyalty program is

)()()()()(XBSdGrkXSNv

XdBdSdGrkNvCS++−−+++=

−+−−+−+=∆µσ

µσ (A3)

Substituting (A2) into (A3),

GcrXSNvCS g )( −−++≥∆ µσ

Note that ,v ,σ and µ are all non-negative. If customers buy when the surplus is

the same, then .0>µ Conversely, if customers do not buy when the surplus from

buying is zero, then .0>σ Together with the assumption that each component of

45

the partition is non-empty, this implies that 0>∆CS for r sufficiently close to gc

and thus the customer will abandon the program.

Imposing additional quantity restrictions on the loyalty program, like

requiring each customer to buy a minimum quantity of both products, lowers the

customer’s surplus from participating in the program. Therefore, the customer

benefits even more from abandoning such programs, and the result still holds.

46

7. References

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Burstein, M. L. (1960). “The Economics of Tie-in Sales.” Review of Economics and Statistics. 42(1): 68-73.

Cairns, R. D. and J. W. Galbraith (1990). “Artificial Compatibility, Barriers to Entry, and Frequent-Flyer Programs.” Canadian Journal of Economics. 23(4): 807-16.

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